Transversal
A pair placement that no line matching, symbol transversal or coupled cover admits is impossible
How to use it
One technique, three passes over the same structure; work them cheapest-first and stop at the first that yields.
- Line matching. Take one row or column and list its unresolved cells with their viable pairs. The line must marry its leftover letters to its leftover digits, every pair distinct: enumerate the marriages; a pair in no marriage is blocked, a cell left with a single marriage is placed.
- Symbol tables. Pick one symbol and write its leftover pairs with the cells that still host them. Three recognitions, cheapest first: a bilocal pair (two hosts) spanning a rectangle kills both diagonals — whichever corner takes the pair, both of its symbols are used there, so the diagonals lose every pairing of either; the two covering rules — block a placement that strands a needed line or wipes a pair, lock a pair into its last home; and, only when the table stays silent, the full enumeration — count the table's completions, block a pair that appears in none, place a pair supported in exactly one cell. Digit tables work identically.
- Dual cover. For one pair, pin a proposed home into both its letter table and its digit table, close both under the covering rules and cross forced cells between them (the two copies of the pinned pair are the only allowed overlap); if the tables cannot meet, the home is impossible.
Shapes worth spotting before any table: the bilocal rectangle above; a unique coverer (one pair can still enter a needed line); a forced cell (a pair with one home left); and the starvation count — a candidate whose placement would leave one of its six houses (its cell, its four symbol lines, the pair itself) optionless. Spotted without any hypothesis, that plain count is rare enough that a forcing single usually catches it first; the starvation-chain argument below is the more common shape, where the starvation only shows up after tentatively placing a candidate.
Why it is sound
Every pass rests on the transversal lemma: in any solution the cells of a fixed symbol form a permutation matrix — one per still-free row and column — carrying every partner exactly once. Line mode: the unresolved cells of one row or column must take distinct letters and distinct digits, so a pair that survives no full matching of the line is impossible. Symbol mode: the two covering rules (a still-needed line with one coverer locks that pair; a pair with one cell takes its row and column from the others) only delete placements no completion uses, so their locks and refutations are sound; the exact pass then tries every completion of the symbol's table and blocks a pair that appears in none — exact, because it checks every possibility. Dual mode: a solution must contain both transversals of a pair's letter and digit, meeting at that pair's single cell, so a home where the two coupled tables cannot meet is impossible. One warning from the theory: choosing one row- and column-disjoint cell per pair is a hard matching problem in general — tractable here only because the grid is at most 9×9, which is why it pays to try the cheap recognitions first and fall back to checking every possibility only when they stay silent.
The seven arguments
Transversal reasoning breaks into seven distinct arguments; a walkthrough names whichever one actually applies on a given step. Each has its own page:
- Line transversal — A row or column's pairs must form a matching of its letters to its digits; forced completions are sound
- Pair coloring — Bilocal pair conjugate links: rectangle diagonals and mono-digit pivot chains eliminate letter/digit options
- Line cover — A pair placement that would leave a row or column of its letter or digit class with no home is impossible
- Starvation chain — A candidate that would leave one of its six houses optionless, directly or through forced single survivors, is impossible
- Transversal chain — Try a candidate pair, propagate forced deductions within one symbol's transversal; if a row/col/digit empties, the candidate is impossible
- Dual-transversal cover — A pair home whose linked letter and digit cover tables cannot meet there is impossible
- Exhaustive transversal count — A pair placement that fits no valid transversal (one cell per row and column) is impossible
A worked example
Take the pair G2: across the whole grid it can only sit at r3c5 or r7c3 — a bilocal pair, and the two hosts span a rectangle. Whichever corner takes G2, both G and 2 are used there, so the diagonals r3c3 and r7c5 lose every pairing of either symbol: G3, G4 and G5 are blocked at r7c5 in one move, and the letter G dies at r7c5 outright. The same walk that found this rectangle runs the line matchings, the per-symbol covering tables with their full transversal counts, and the coupled dual tables — cheapest recognition first, exact enumeration only when the cheap passes are silent.